Eigenvalue theory and perturbation

Reading: Golub & Van Loan §7.1–7.2, pp. 348–365.

By the end of this week you should be able to

• Use similarity transformations and state the Schur decomposition.
• Decide when a matrix is diagonalizable and what to do when it is not.
• Bound eigenvalue perturbations with the Bauer-Fike theorem.

Algorithms introduced

• Schur decomposition
• Eigenvalue condition numbers

Where this shows up in AI

The spectral radius governs whether an iterated map converges or diverges. Nonsymmetric spectra can be extremely sensitive, which matters for recurrent dynamics.

Materials